What is the difference between the 'Symmetry Mirror' and 'X Mirror' options when weight painting?
I can get you part of the way there, but I have the same question as you with regard to the ones I don't know. Any further input is therefore very welcome.

The Weight is the setting which controls the degree of influence that your brush will have in assigning vertex weight values. Setting to .5, say, would mean that you want the vertex weight values that will be added via weight painting to be able to reach a value as high as .5, but never go higher. It sets the upper-most limit for the weight values, so that you will never get higher ones from  painting.

The Radius setting is simply the size of your brush.

The Strength is the setting which determines the rate at which an applied brush stroke will have whatever affect it will have, as determined by other settings. The higher it is, the less duration is needed (and less pressure if you use a tablet) before its full effect can be finally realized.

The Falloff: Cursor has to do with how the brush's effect will evaluate in space. There is the ability to have a brush which only affects the surface of an object, in a kind of 2D painted-on way, or alternatively, it can affect the object as if the brush itself were a 3D sphere, affecting anything within its zone of influence. As to which of these two behaviors comes along with the Cursor option, I'm not sure.

The Mirror option under the Symmetry section is one I definitely know about, as it caused me grief in the beginning. What it does is mirror painted weights. However, it doesn't copy vertex weight data to any other vertex groups. Instead, it copies vertex weight data to the other side of the mesh, but that data will still be part of the currently targeted vertex group. Meaning that you will get influence from whatever bone corresponds to the vertex group you are currently targeting not only on this side of the mesh, where you want it, but on the other side of the mesh also, where you almost certainly don't want it. I don't use this option, and I don't know why anyone would.

Auto-Normalize, I have to guess about a bit. I think it is possible to have all your vertex weights add up to a value of 1, meaning that any changes in weight painting are only possible by altering the ratios of all weights, such that they will all still add up to one in total, no matter how they are changed. My guess is that this setting allows normalization to be an "always on" feature. But I'm not an expert on this. I don't yet know why normalization is even used, nor why it is useful. I'm still trying to wrap my head around it. Sorry I can't tell you more.

Now, X-Mirror is almost certainly what you actually wanted when you first selected Mirror under the Symmetry section. It does what you usually want from symmetry. It copies the vertex weight data from the currently targeted vertex group and adds it to the corresponding vertex group on the other side of the mesh. This is a real time saver, as it means you can mostly only weight paint one side of your mesh, and get mirrored weight values automatically. It only works if you have corresponding vertex groups already in existence, of course. But you likely will, since most folks always bind their meshes using the With Automatic Weights option.

Topology Mirror is the one setting that I have absolutely no clue about. Nothing I have read, stumbled upon, or actively researched has helped me get any idea what it is, or when I would use it. I feel like I should at least have a word or two on the subject by now. But I don't. Not even after reading the Blender manual. So I'm hoping someone can help us both out here.

That's all I know. Hope it helps.